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A Lower Bound on Swap Regret in Extensive-Form Games

Constantinos Daskalakis, Gabriele Farina, Noah Golowich, Tuomas Sandholm, Brian Hu Zhang

TL;DR

The paper establishes an information-theoretic lower bound showing that no swap-regret minimization algorithm for general extensive-form games can run in polynomial time in the game size and 1/ε. By embedding a hard normal-form adversary into a tree-form decision problem and leveraging concentration properties, the authors prove that achieving average swap regret ε requires at least exp(Ω(min{ m^{1/14}, ε^{-1/6} })) rounds, thereby ruling out poly(m,1/ε) algorithms. This result closes a long-standing gap between swap-regret bounds in normal-form and extensive-form settings and demonstrates an exponential separation between swap-regret and weaker regret notions in EF games. The findings have implications for the computability of NFCEs via learning dynamics and suggest fundamental limits on efficiently computing equilibrium concepts in extensive-form games.

Abstract

Recent simultaneous works by Peng and Rubinstein [2024] and Dagan et al. [2024] have demonstrated the existence of a no-swap-regret learning algorithm that can reach $ε$ average swap regret against an adversary in any extensive-form game within $m^{\tilde{\mathcal O}(1/ε)}$ rounds, where $m$ is the number of nodes in the game tree. However, the question of whether a $\mathrm{poly}(m, 1/ε)$-round algorithm could exist remained open. In this paper, we show a lower bound that precludes the existence of such an algorithm. In particular, we show that achieving average swap regret $ε$ against an oblivious adversary in general extensive-form games requires at least $\mathrm{exp}\left(Ω\left(\min\left\{m^{1/14}, ε^{-1/6}\right\}\right)\right)$ rounds.

A Lower Bound on Swap Regret in Extensive-Form Games

TL;DR

The paper establishes an information-theoretic lower bound showing that no swap-regret minimization algorithm for general extensive-form games can run in polynomial time in the game size and 1/ε. By embedding a hard normal-form adversary into a tree-form decision problem and leveraging concentration properties, the authors prove that achieving average swap regret ε requires at least exp(Ω(min{ m^{1/14}, ε^{-1/6} })) rounds, thereby ruling out poly(m,1/ε) algorithms. This result closes a long-standing gap between swap-regret bounds in normal-form and extensive-form settings and demonstrates an exponential separation between swap-regret and weaker regret notions in EF games. The findings have implications for the computability of NFCEs via learning dynamics and suggest fundamental limits on efficiently computing equilibrium concepts in extensive-form games.

Abstract

Recent simultaneous works by Peng and Rubinstein [2024] and Dagan et al. [2024] have demonstrated the existence of a no-swap-regret learning algorithm that can reach average swap regret against an adversary in any extensive-form game within rounds, where is the number of nodes in the game tree. However, the question of whether a -round algorithm could exist remained open. In this paper, we show a lower bound that precludes the existence of such an algorithm. In particular, we show that achieving average swap regret against an oblivious adversary in general extensive-form games requires at least rounds.
Paper Structure (12 sections, 11 theorems, 12 equations, 1 figure, 1 table)

This paper contains 12 sections, 11 theorems, 12 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. No-regret learning and $\Phi$-regret.
  4. Tree-form decision problems and extensive-form games.
  5. Previous Swap Regret Minimization Algorithms
  6. Normal-form games
  7. Extensive-form games and tree-form strategy sets
  8. Main Result
  9. Extensive-form game instances.
  10. The adversarial environment.
  11. Analysis.
  12. Conclusion and Future Research

Key Result

Theorem 1.1

There is no swap regret minimization algorithm for general extensive-form games that requires fewer than $\exp(\Omega\qty(\min\qty{m^{1/14}, \epsilon^{-1/6}}))$ rounds.

Figures (1)

  • Figure 1: A depiction of the class of tree-form decision problems used in the proof of \ref{['th:efg-lower']}. Triangles ($\color{p1color}\blacktriangle$) are decision points and boxes ($\square$) are observation points.

Theorems & Definitions (14)

  • Theorem 1.1: Main theorem, informal
  • Proposition 2.1
  • Theorem 3.1: Blum07:External
  • Theorem 3.2: Dagan23:ExternalPeng23:Fast, upper bound
  • Theorem 3.3: Theorem 4.1 of Dagan23:External, lower bound
  • Corollary 3.4: Dagan23:ExternalPeng23:Fast, tree-form upper bound
  • Theorem 4.1: Main theorem
  • Theorem 4.2: Dagan23:External, expanded version of \ref{['th:lb']}
  • Lemma 4.3
  • proof
  • ...and 4 more